Minimum overshoot design for SISO discrete-time systems

The problem of designing a controller to minimize the maximum negative error due to a specific bounded input is studied. It is found that the solution can be obtained by solving an infinite linear programming problem. In the special case of a step input, a simple analytical formula for minimum overshoot is derived for a plant with one unstable pole and one nonminimum phase zero. The solution technique has potential application to other control problems, which seek to minimize quantities not directly representable as a norm